The chamber complex for the Littlewood-Richardson coefficients of GL4

THE CHAMBER COMPLEX FOR THE LITTLEWOOD-RICHARDSON COEFFICIENTS OF GL4. EMMANUEL BRIAND, MERCEDES ROSAS, AND STEFAN TRANDAFIR The Littlewood-Richardson coefficients cν λ,µ are a family of constants of central importance in representation theory. They are indexed by triples of integer partitions. It is known that when the size of the partitions is limited to some fixed integer N(coefficients associated to GLN), these coefficients are given by piecewise polynomial formulas in λ1,...λN, µ1, . . . , µN, ν1, . . . , νN−1. More precisely, the domains of polynomiality are the big cells (”chambers”) of a subdivision of a cone in R3N−1(the ”chamber complex”). In 2005, ´ E. Rassart has calculated explicitly the chamber complex for N= 3, finding 18 chambers subdividing a cone in R8, and the corresponding polynomial formulas (in this case, linear). Using Rassart’s description, E. Briand and M. Rosas recently listed exhaustively all symmetries of the chamber complex for N= 3, finding a group of 288 symmetries (much more that the 24 symmetries known to exist for general N). It is natural to look at what happens in the next case, N= 4. With this aim, we have computed (using Sagemath and Singular) the chamber complex for N= 4. This object is large, since it consists in the subdivision of a cone in R11 in 67769 chambers. This computation is based on known combinatorial descriptions of the Littlewood-Richardson coefficients. For instance, the Littlewood-Richardson coefficient c(26,21,14,11) (16,13,5,2),(17,12,6,1) counts the triangles 36 34 53 29 x365 16 x1x271 0 26 47 61 72 filled with integers subject to the inequalities a+b≥c+din all rhombi: d b c a b c b d a c a d of the triangle. (The bottom border entries 0,26,47,61,72 are the cumulated sums of (26,21,14,11), that are: 0,26,26+21,26+21+14,26+21+14+11 and the top border entries 0,16,29,34,36,5365,71,72 are the cumulated sums of (16,13,5,2) and (17,12,6,1), that are 0,16,16+13,16+13+5,...,16+13+5+2+17+12+6+1). E. Briand, Departamento Matem´ atica Aplicada I, Universidad de Sevilla M. Rosas, Departamento de ´ Algebra, Universidad de Sevilla S. Trandafir, Simon Fraser University. E. Briand and M.Rosas are partially supported by MTM2016-75024-P and FEDER, and Junta de Andalucia under grant FQM333.